MTH641 GDB Solution 1 Fall 2020-2021

MTH641 GDB Solution Fall 2020-2021 Solution

We are interested in the study of the existence of continuous solutions of the following nonlin

Fredholm integral equation,

x(t) =f (t) + ò g(t, s, x(s))ds, -¥ < a £ t £ b < +¥ (1.1

 Where f (t) ÎC ([a,b]) . Usually the proof of the existence of a solution of (1.1) starts with some conditions on the function g (t, s, x) as well as the limits of integration a,b and the function f (t). Based on these conditions, a Banach space is chosen in such a way that the existence problem is converted into a fixed-point problem for an operator avers this Banach space.

In the first case, we use some conditions on the function g(t,s, x) and we require that g(t,s, x) be bounded w.r.t. x. Then, we use Schaefer’s fixed-point theorem and prove the existence of a solution belonging to C([a,b]). In the second case, we replace the strong condition that g(t,s, x) is bounded w.r.t. x by a weaker condition. To prove the existence of a continuous solution of (1.1) in this case, we introduce a new norm

f (t) m Over the space C ([a, b]) and use Schauder’s fixed-point theorem.

In the second part of this work, we study the following nonlinear Volterra equation,

x(t) =tf (t) + ò g(t, s, x(s))ds = f (t) + Tx(t), -¥ < a £ t £ b < +¥ (1.2)

a

Where

f (t) ÎC ([a,b])

The main tool in the proof of the existence of a solution of ( 1.2) is the Leray—Schauder principle combined with Gronwall’s inequality. Also, we prove the uniqueness of the solution of (1.2) by showing that there exists an n Î N such that T n is a contraction on some closed ball of C([a,b]) containing the possible solutions of (1.2). we prove the existence of continuous solutions of (1.1). In the second part, we investigate the existence and uniqueness of the solution of the nonlinear Volterra equation (1.2)

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